What is the integral of y/(1+x^2y^2) ?

The integral of y/(1+x^2y^2) is arctan(yx)+C

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integral of y/(1+x^2y^2)

\int \frac{y}{1 + x ^{2} y ^{2}} d x

arctan (y x) + C

Solution steps

\int \frac{y}{1 + x ^{2} y ^{2}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= y \cdot \int \frac{1}{1 + y ^{2} x ^{2}} d x

Apply Integral Substitution

= y \cdot \int \frac{1}{y ( 1 + u ^{2} )} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= y \frac{1}{y} \cdot \int \frac{1}{1 + u ^{2}} d u

Use the common integral:

\int \frac{1}{1 + u ^{2}} d u = arctan (u)

= y \frac{1}{y} arctan (u)

Substitute back

u = y x

= y \frac{1}{y} arctan (y x)

Simplify

y \frac{1}{y} arctan (y x) : arctan (y x)

= arctan (y x)

Add a constant to the solution

= arctan (y x) + C

\int \frac{- cos ( n x )}{n} d x

t \frac{d y}{d t} = t^{3} + 24 t^{3} y

\frac{\partial}{\partial x} (7 y cos (x))

n = 2 \sum \infty \frac{5}{3 ^{n}}

- \frac{32}{( t ) ^{3}} + \frac{4}{( t ) ^{2}}

What is the integral of y/(1+x^2y^2) ?
The integral of y/(1+x^2y^2) is arctan(yx)+C